It was proposed in [1]} that the soliton gas can be described by the thermodynamic limit of the spectral \(N\) -gap KdV solutions, achieved by assuming a special band-gap distribution (scaling) of the spectral set \(\mathcal {S}_N\)  for \(N \rightarrow \infty \)  on a fixed interval (say \(\lambda \in [-1, 0]\) ) whereby the finite spectral bands are required to be exponentially narrow compared to the gaps \(\sim 1/N\)  between them. Such spectral scaling enables the soliton limit of finite-gap potentials as \(N \rightarrow \infty \)  while preserving finite spectral DOS.
